whenever the right hand side makes sense and defines an analytic function in \(\mathbb {D}\) . The operators \(\mathcal {H}_{\mu ,1}\)  have been extensively studied in [1]}, [2]}, [3]}, [4]}, [5]}.
In this case, if we let \(\mu \)  be the Lebesgue measure on \([0,1)\) , we can find that \(\mathcal {H}_{\mu ,1}\)  is just the classical Hilbert matrix \(\mathcal {H}=\big ((n+k+1)^{-1}\big )\) , which induces the classical Hilbert operator (see [6]}, [7]}, [8]} for more details). For the case \(\alpha =2\) , we have studied the operator in [9]}, [10]} and we call \(\mathcal {H}_{\mu ,2}\)  the Derivative-Hilbert operator. In this paper, we also call \(\mathcal {H}_{\mu ,\alpha }\)  \((\alpha >0)\)  the generalized Hilbert operator.
